Vega, Volatility, and Why Long-Dated Options Are All Vol

Completing the set: where gamma and theta die, vega is what is left standing.

The intuition

The argument in one breath:

more vol ⇒ the option is worth more ⇒ vega is exactly that sensitivity ⇒ a long option is always long vega.

Vega measures how much you make when implied volatility rises. Buy any option — call or put — and you are long vega: a higher vol input lifts the price. (Calls and puts have the same vega, because put–call parity, $C-P = S-Ke^{-r\tau}$, has no $\sigma$ in it, so $\partial C/\partial\sigma = \partial P/\partial\sigma$.)

The formula, read two ways

With $r=q=0$,

$$\nu \;=\; \frac{\partial V}{\partial\sigma} \;=\; S\sqrt{\tau}\,\varphi(d_1),\qquad \varphi(x)=\tfrac{1}{\sqrt{2\pi}}e^{-x^2/2}.$$

It says two things a trader lives by:

• A maturity story ($\sqrt{\tau}$): the longer the option has to live, the more a change in vol compounds into its price. Short-dated options barely have vega; long-dated ones are dominated by it.
• An at-the-money story ($\varphi(d_1)$): vega is largest near the money and fades for deep in- or out-of-the-money strikes. Vol risk lives around the strike.

The numbers — vega is a maturity story

At-the-money vega grows with $\sqrt{\tau}$ ($S=100$, $\sigma=20\%$, $r=q=0$; vega per $1.00$ of $\sigma$):

(Per one vol point, divide by 100: a one-year ATM option gains $\approx 0.40$ per share for each $+1\%$ of implied vol.) A five-year option carries roughly eight times the vega of a one-month one — same underlying, same strike.

The payoff: where gamma and theta die, vega dominates

This is why the Greeks split by maturity. Compare vega with the two Greeks from the earlier notes:

$$\frac{\nu}{\Gamma} \;=\; S^{2}\sigma\tau, \qquad\qquad \frac{\nu}{|\Theta|} \;=\; \frac{2\tau}{\sigma}.$$

Both grow with maturity. So as an option lengthens, its vega dwarfs its gamma and its theta:

• Short-dated options are a gamma/theta world: a tall gamma spike at the strike, fast decay, almost no vega. You trade realised vol, day by day.
• Long-dated options are a vega world: gamma and theta have flattened to almost nothing, and the position is essentially a bet on the level of implied vol.

That is the sense in which "vega takes over" once gamma and theta fade.

The honest caveat

Vega is not immortal either. In absolute terms it also $\to 0$ as $\sigma\to\infty$ (the same $\varphi(d_1)\to 0$), and at-the-money vega actually declines gently once vol is very high — once a call is already worth almost $S$, extra vol can add little. But vega carries no $1/\sigma$ and no $\sigma^2$ factor, so it fades the slowest of the three — which is exactly why it is what remains when gamma and theta have collapsed.

Takeaway. Vega, $\nu = S\sqrt{\tau}\,\varphi(d_1)$, is your exposure to the level of implied vol — always positive for a long option, largest at-the-money, and above all a maturity story. Short-dated: gamma and theta rule. Long-dated: vega rules. $\nu/\Gamma = S^2\sigma\tau$ and $\nu/|\Theta| = 2\tau/\sigma$ both grow with time.

Assumes $r=q=0$ and a flat volatility surface (the exam convention); $S=100$ in the table, vega quoted per unit of $\sigma$ (divide by 100 for a one-point move). Completes the set: delta (direction), gamma (convexity), theta (the rent), vega (the vol level).